Indirect Measurement Technique: Using Trigonometric Ratios Grade Nine


 Sharyl Nichols
 5 years ago
 Views:
Transcription
1 Ohio Standards Connections Measurement Benchmark D Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates. Indicator 4 Use scale drawings and right triangle trigonometry to solve problems that include unknown distances and angle measures. Mathematical Processes Benchmark A Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution. Related Standards Measurement Benchmark C Apply indirect measurement techniques, tools and formulas, as appropriate, to find perimeter, circumference and area of circles, triangles, quadrilaterals and composite shapes, and to find volume of prisms, cylinders, and pyramids. Lesson Summary: Students will find the height of an object that would be difficult or impossible to measure directly. They will construct and use a clinometer to measure the angle of elevation (or depression). Students will create a sketch of the measurement situation using a right triangle, indicate relevant points in the physical situation, and use basic trigonometric ratios to calculate the desired lengths. Estimated Duration: 90 minutes (excluding pre and postassessments). Commentary: In this lesson, students will apply basic right triangle trigonometric ratios in order to determine the height of an object. They will use a homemade clinometer. There are many important concepts embedded in this activity. First, there is no obvious right triangle present. Students must be able to visualize an appropriate right triangle and determine what locations are its vertices. Second, it is not immediately obvious to many students exactly what angle is being measured by the clinometer and how that angle is related to the angle of elevation to the height of the object. Third, students must understand the function of the weighted string and how it is related to the other "angles" and "sides" present in the experiment. Another aspect of this problem is the ability to create an equation from the problem situation and solve that equation. Before completing the measurement activity, students should have ample time to experiment with the clinometer to determine what angle they are measuring and what other parts of the triangle are measurable. PreAssessment: Students should be provided a problem set that enables them to demonstrate their understanding of the basic trigonometric ratios, sine, cosine and tangent. Emphasis should be given to the tangent ratio and distinguishing the difference between the adjacent side and opposite side with respect to a given angle. 1
2 Geometry and Spatial Sense Benchmark H Establish the validity of conjectures about geometric objects, their properties and relationships by counterexample, inductive and deductive reasoning, and critiquing arguments made by others. Benchmark I Use right triangle trigonometric relationships to determine lengths and angle measures. Use PreAssessment: Indirect Measurement, Attachment A, or select problems from your curriculum that effectively asses the topic. Invite students to share their solutions to each of the three questions on the preassessment and explain their thinking. Allow students the opportunity to correct any errors presented. Ask appropriate followup questions to ensure that students understand the basics of trigonometric ratios. Instructional Tip: Walk around the room while students are working and record several of the students answers for the tan A. When students have completed the preassessment, write the collection of answers you recorded on the board and challenge students to determine which is correct. Have students provide justification for the correct solution. Then, engage students to determine what might have led to the incorrect answers. Continue this activity until students seem to understand how to select the appropriate sides to find tan A. During informal observation of student work, look for students having difficulty with sine or cosine. If needed, discuss the other ratios using a process similar to that described above. Scoring Guidelines: Informally evaluate students strengths and weaknesses. Walk around the room and observe the progress and take notes as needed. Instructional Tip: This assessment was designed to be an instructional review. All or most students struggle with a specific part of the preassessment may indicate the need for a minilesson focused on that specific topic prior to continuing with this lesson. Some of the more prevalent errors students encounter includes the following: Not remembering the side relationships for each ratio. Not realizing that the side designations are related to a specific angle (Is this always challenging or only for angles in certain positions or when triangles are rotated in a different orientation? Note the orientation of the triangle used on the preassessment.) 2
3 Not remembering which side is opposite or adjacent (i.e., unfamiliar terminology can be hard to recall.) Difficulty performing necessary computations or algebraic simplification. PostAssessment: Students will describe another situation where this technique could be used to measure an object from their own environment. Then, they will use the clinometer and this technique to determine the size (height, length, etc.) of the object chosen. The object they select should not be something measured during this lesson. Students should provide a written explanation of how they set up the situation to apply the technique. Include explanations for finding relevant angles (angle of elevation or depression) and measuring necessary lengths. Students should support their description and explanation with labeled sketches to show the situation including all pertinent objects, measures and variables. Scoring Guidelines: The postassessment will be scored using a rubric. Students should be provided with a copy of the rubric prior to the assessment. Make sure students understand the assignment as well as all aspects of the scoring rubric prior to beginning the assessment. Instructional Tip: It is suggested that you set aside time with your students to discuss the parameters of the assessment prior to assigning it. You may want to consider making adjustments to the assignment or rubric based upon input from the students. The last column of the rubric was left blank intentionally to encourage student teacher negotiated additions. Sample Analytic Scoring Rubric: Score Situation Explanation of setup and execution 4 Clearly stated and accurate 3 Clearly stated, but not very reasonable Understanding of measurement technique evident One or more minor concepts missing or not mentioned Labeled sketch Sketch with complete labeling and scaled reasonably accurate Sketch with most things labeled and scaled adequately Other Criteria 3
4 Score Situation Explanation of setup and execution 2 Cleary stated but copied from class or HW 1 Not clearly stated 0 Not included or copied from class or HW One key concept missing An attempt is made in written format No evidence demonstrated or no attempt Labeled sketch Sketch with missing labels and poorly scaled Sketch attempted, but hinders correct solution Not included or copied from class or HW Other Criteria Instructional Procedures: 1. Prior to the lesson, select an object to be measured. Some possibilities include the height of a flagpole or the school building, the height of a basketball backboard or a marked height on a wall like the top of a window in your classroom. The object should be tall enough to make it difficult to measure directly. 2. Create one clinometer in advance using the directions and the diagram provided below: Tape one end of a 6inch string to the midpoint (directly between the 0 o and the 180 o mark on the protractor) on the straight side of the protractor. Tape two pieces of string across the diameter on one end of the straw forming a crosshair. Tape the straw to the straight side of the protractor. Tie the hex nut (or weight) to the other end of the string. 3. Inform students that a clinometer is a measuring device and show them the premade model. Demonstrate looking through the end of the straw (without the crosshair) and 4
5 getting the angle measurement (wait for the string to stop swaying and then pinch the string against the curved edge of the protractor). 4. Distribute Using a Clinometer, Attachment B, with the directions for building a clinometer. Organize students into pairs and instruct them to use the directions and the premade model to build a clinometer. Students should work in pairs to practice sighting objects either above or below their line of sight. 5. Encourage students to experiment with the clinometers, by asking them questions similar to the following: What angle does the clinometer measure with respect to what you are looking at through the sight? How would you define "angle of elevation" or "angle of depression" as it relates to the objects you are sighting? How is the angle of elevation (depression) related to the angle measurement where the string crosses the scale of the protractor? Identify a right triangle that includes the object and the angle of elevation (depression)? What are the other parts (sides/angles) of that triangle? What parts of your triangle could you actually measure? How could those parts be used to determine the measurements you want to know? 6. Facilitate a whole class discussion to share observations and to review answers to the questions used for the clinometer exploration. Instructional Tip: The next step describes a whole class demonstration. Some students may be ready to work in groups independently with a little guidance. Determine the best strategy for your students and their understanding. 7. Inform students they will help you to measure an object (point out or describe whatever you preselected to measure) using a clinometer. For example, if you selected to measure the distance from the floor to an "X" on a wall the following would apply: Place an X on the wall with masking tape (for a whole class demonstration, this should be done prior to class). Select a student to sit in a chair 10 feet or 120 inches from the wall (measure from the wall to the student s eyes). The student will use the clinometer to measure the angle of elevation by aligning the crosshair with the center of the "X" on the wall. Measure the height from the floor to the student s eyes. Ask students why this measurement is needed. (Note: This measurement will be added to the solution of the trigonometry problem.) Ask the students to label the diagram of the situation using the sketch provided on Using a Clinometer, Attachment B, and shown below. Students may be encouraged to work with a partner for this portion, as needed. 5
6 After waiting sufficiently for students to think about the situation, select a student to share his/her labeled sketch. Discuss as a class until a consensus is reached and a correctly labeled sketch is completed. Include in your discussion how to find the angle of elevation. Refer to the diagram below for assistance. Ask students about the different 90 angles and to explain why. <BAC <HAZ Note: <ZAB = 90 0 <HAC =
7 Ask students to consider if trigonometric ratios can be used to help find BE (distance from the point on the wall to the floor in the problem situation.) If so, which ratio and how? If not, why not? Provide sufficient time for students to consider the problem and then discuss the solution as a class. If all goes well, students should provide most of the input during the discussion. Solution for Class Demonstration: AC was measured as 10 feet (120 inches) from the student s eyes to the wall and AD was measured (we are using 36 0 for the sample). BAC was found using the clinometer and should be about BE can be found as follows: tan 10 0 = 120 tan 10 0 BC = 120 BC = 21 inches then to get BE: BE = BC + AD BE = BE = 57 inches BE = 4' 9" 8. Organize the students into groups of two or three and provide additional practice using this technique. They can measure things around the school building or outside. Challenge the students to find situations to measure that would involve an angle of depression (i.e., something down stairs or down hill) or to measure something that might involve a different trigonometric ratio (i.e., you can measure the length of the hill but cannot measure an object directly in front of you). Be prepared to offer suggestions for measuring situations. Instructional Tip: Optionally provide students with a list of measurement situations from around the school grounds. Then allow students to select from the list. This may be useful to keep students engaged and on task. 9. Discuss objects measured, strategies used for measuring, challenges encountered, and solutions. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). 7
8 Students may work in collaborative groups to complete this lesson. It may be beneficial to have heterogeneous groups to enable peer tutoring during the activities. Provide students with predrawn diagrams and let the students record the measures (refer to Using A Clinometer, Attachment B. Consider using the Indirect Measurement Techniques Grade Eight lesson with students who are not ready to work with the trigonometric ratios. Challenge the students to use the clinometer to measure objects involving an angle of depression instead of an angle of elevation; e.g., measure something that is situated below eye level. Challenge the students to use the clinometer to measure objects involving the other trigonometric ratios, sine or cosine; e.g., if there were a tree situated at a bottom of a hill, then a student could measure the length of the hill using a tape measure and walking the hill. Then use the cosine to find the distance from a person standing on the top of the hill to the top of the tree or use the sine to find the height of the tree. Extensions: These are ideas for all students to continue learning on this topic  in the classroom or outside of the classroom. Select three indirect measurement situations for the students to use their clinometers outside the school that would be difficult to measure directly. Each measurement situation will be a station. Divide the class into three groups (Groups 1, 2 & 3). Each group will be given time to collect data (i.e., agree on the angle measure using their clinometers), measure the distance between the clinometer and the ground and to find any other measures needed. Then, groups should discuss setup (i.e., whether to use an angle of elevation or depression and which trigonometric ratio should be used to find the desired measurement). Groups will rotate through the three stations and collect data and discuss setup at each station. Complete the calculations and create sketches outside of class (optional). Each student should create a scale drawing and show the work needed to calculate the specified measure for each station. Discuss solutions for each station within the three groups in class. The larger groups need to reach consensus, if possible, or at a minimum agree on a final answer. Prepare for students to do a jigsaw discussion about how their groups performed the indirect measurements at each of the three stations and to share their group s findings: a. Organize the students into groups of three students with at least one student from each group. b. The students will take turns discussing the process or steps their group used to find the measures at each station. Students should discuss challenges and how their group worked through them. 8
9 Students should reconcile differences in their solutions. For example, each group probably stood at different distances from the object, but the answers should be the same or close. Why are the answers not different? They should conclude that the distance from the object does not affect the final solution. Circulate the room and provide assistance as needed and keep the student discussions on task and moving so they can progress through to consider objects 2 and 3. Refrain from providing the students with the right answers. Home Connections: Each student must choose an object near his/her home and find the height using the clinometer. A written description of the situation a scale drawing and work must accompany the solution. Resources for parent and family involvement are available on the Internet. Interdisciplinary Connections: Environmental and Agricultural Sciences  This indirect measurement technique could be used to find the height of trees. The solution could be used when determining the amount of lumber or firewood a tree would likely produce. Geography  This indirect measurement technique could be used to find the height of natural landmarks such as mountains and caverns; e.g., cartographers use similar techniques when making maps. Key Vocabulary: angle of depression angle of elevation basic trigonometric ratios clinometer cosine (cos) indirect measurement sine (sin) tangent (tan) Technology Connections: A calculator with trigonometric functions will make the lesson more manageable and eliminate the need for looking up values in tables. Interactive calculators with trigonometric functions are available on the Internet. Students can use word processing and productivity software to report their findings visually and in writing. Students could use a database to input their data for the class to view and compare results. Interactive charts could also be created from the collected data. 9
10 Materials/Resources Needed: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: Attachment A, PreAssessment: Indirect Measurement, Attachment B, Indirect Measurement Using A Clinometer, a preassembled clinometer to show to students, parts needed to assemble a clinometer (transparent tape, a small spool of thread, drinking straw, protractor, string, hex nut or weighting device), tape measure, masking tape. For the students: Parts needed to assemble a clinometer (transparent tape, thread, drinking straw, protractor, string, hex nut), tape measure, masking tape. Attachments: Attachment A, PreAssessment Indirect Measurement Attachment B, Using a Clinometer 10
11 Attachment A PreAssessment: Indirect Measurement Name: Date: Complete the following exercises in any order that you would like. Give the basic trigonometric ratios for the ABC. sin A = sin B = cos A = cos B = tan A = tan B = Label the sides of the following triangle with respect to opposite adjacent hypotenuse T using the following: Use the labels to write the basic trigonometric ratios for T below. 11
12 Attachment B Using a Clinometer Name: Date: Materials needed to build a clinometer: protractor, drinking straw, transparent tape, string, thread, and a hex nut. Follow the directions below to construct a clinometer: Tape one end of a 6inch string to the midpoint (directly between the 0 and the 180 mark on the protractor) on the straight side of the protractor. Tape two pieces of string across the diameter on one end of the straw forming a crosshair. Tape the straw to the straight side of the protractor. Tie the hex nut to the other end of the string. To use the clinometer you look through the end of the straw (without the crosshair) and align the crosshair with the object, allow the string to stop swaying and pinch the string against the curved edge of the protractor. The place where the string crosses the angle measures on the protractor will indicate the angle of elevation or the angle of depression. A sketch of the situation is show below. Label the things in the sketch including known measurements. Use variables to represent the unknown measurements. (Do not forget to label the angle represented by that shown on the clinometer.) 12
Similar Triangles Grade Seven
Ohio Standards Connection Geometry and Spatial Sense Benchmark E Use proportions to express relationships among corresponding parts of similar figures. Indicator 1 Use proportional reasoning to describe
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationDrawing Lines of Symmetry Grade Three
Ohio Standards Connection Geometry and Spatial Sense Benchmark H Identify and describe line and rotational symmetry in twodimensional shapes and designs. Indicator 4 Draw lines of symmetry to verify symmetrical
More informationG C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
Performance Assessment Task Circle and Squares Grade 10 This task challenges a student to analyze characteristics of 2 dimensional shapes to develop mathematical arguments about geometric relationships.
More informationGeometric Transformations Grade Four
Ohio Standards Connection Geometry and Spatial Sense Benchmark I Describe, identify and model reflections, rotations and translations, using physical materials. Indicator 7 Identify, describe and use reflections
More informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationWeek 1 Chapter 1: Fundamentals of Geometry. Week 2 Chapter 1: Fundamentals of Geometry. Week 3 Chapter 1: Fundamentals of Geometry Chapter 1 Test
Thinkwell s Homeschool Geometry Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Geometry! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson plan
More informationGeometry Notes PERIMETER AND AREA
Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter
More informationGEOMETRY CONCEPT MAP. Suggested Sequence:
CONCEPT MAP GEOMETRY August 2011 Suggested Sequence: 1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons
More informationCommutative Property Grade One
Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using
More informationDiscovering Math: Exploring Geometry Teacher s Guide
Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional
More informationASSESSSMENT TASK OVERVIEW & PURPOSE:
Developing a Trigonometry Phone App I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this activity, students will be asked to develop a program for a smartphone application that could be used to calculate the
More informationInvestigating Quadrilaterals Grade Four
Ohio Standards Connection Geometry and Spatial Sense Benchmark A Provide rationale for groupings and comparisons of twodimensional figures and threedimensional objects. Indicator 3 Identify similarities
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationGlencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 33, 58 84, 87 16, 49
Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 68 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,
More informationNew York State Student Learning Objective: Regents Geometry
New York State Student Learning Objective: Regents Geometry All SLOs MUST include the following basic components: Population These are the students assigned to the course section(s) in this SLO all students
More informationAlgebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard
Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express
More informationLocal Government and Leaders Grade Three
Ohio Standards Connection: Government Benchmark A Identify the responsibilities of the branches of the U.S. government and explain why they are necessary. Indicator 2 Explain the structure of local governments
More informationProblem of the Month: Cutting a Cube
Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:
More informationOverview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.3 Finding Volume of Cones, Cylinders, and Spheres
Cylinders, and Spheres Number of instruction days: 6 8 Overview Content to Be Learned Evaluate the cube root of small perfect cubes. Simplify problems using the formulas for the volumes of cones, cylinders,
More informationNEW MEXICO Grade 6 MATHEMATICS STANDARDS
PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical
More informationLesson Plan Teacher: G Johnson Date: September 20, 2012.
Lesson Plan Teacher: G Johnson Date: September 20, 2012. Subject: Mathematics Class: 11L Unit: Trigonometry Duration: 1hr: 40mins Topic: Using Pythagoras Theorem to solve trigonometrical problems Previous
More informationGeometry Solve real life and mathematical problems involving angle measure, area, surface area and volume.
Performance Assessment Task Pizza Crusts Grade 7 This task challenges a student to calculate area and perimeters of squares and rectangles and find circumference and area of a circle. Students must find
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationThe Primary Trigonometric Ratios Word Problems
The Primary Trigonometric Ratios Word Problems. etermining the measures of the sides and angles of right triangles using the primary ratios When we want to measure the height of an inaccessible object
More information8 th Grade Task 2 Rugs
8 th Grade Task 2 Rugs Student Task Core Idea 4 Geometry and Measurement Find perimeters of shapes. Use Pythagorean theorem to find side lengths. Apply appropriate techniques, tools and formulas to determine
More informationCGE 3b 2 What s My Ratio? The Investigate the three primary trigonometric ratios for rightangled MT2.01 triangles. Summarize investigations.
Unit 2 Trigonometry Lesson Outline Grade 10 Applied BIG PICTURE Students will: investigate the relationships involved in rightangled triangles to the primary trigonometric ratios, connecting the ratios
More informationPlotting Ordered Pairs on a Four Quadrant Grid Grade Five
Ohio Standards Connection Geometry and Spatial Sense Benchmark C Specify locations and plot ordered pairs on a coordinate plane. Indicator 6 Extend understanding of coordinate system to include points
More informationParallel and Perpendicular. We show a small box in one of the angles to show that the lines are perpendicular.
CONDENSED L E S S O N. Parallel and Perpendicular In this lesson you will learn the meaning of parallel and perpendicular discover how the slopes of parallel and perpendicular lines are related use slopes
More informationLinear, Square and Cubic Units Grade Five
Ohio Standards Connection Measurement Benchmark F Analyze and explain what happens to area and perimeter or surface area and volume when the dimensions of an object are changed. Indicator 4 Demonstrate
More informationUnderstanding Ratios Grade Five
Ohio Standards Connection: Number, Number Sense and Operations Standard Benchmark B Use models and pictures to relate concepts of ratio, proportion and percent. Indicator 1 Use models and visual representation
More informationUnit 8 Angles, 2D and 3D shapes, perimeter and area
Unit 8 Angles, 2D and 3D shapes, perimeter and area Five daily lessons Year 6 Spring term Recognise and estimate angles. Use a protractor to measure and draw acute and obtuse angles to Page 111 the nearest
More informationRight Triangles 4 A = 144 A = 16 12 5 A = 64
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
More informationWarning! Construction Zone: Building Solids from Nets
Brief Overview: Warning! Construction Zone: Building Solids from Nets In this unit the students will be examining and defining attributes of solids and their nets. The students will be expected to have
More informationRIGHT TRIANGLE TRIGONOMETRY
RIGHT TRIANGLE TRIGONOMETRY The word Trigonometry can be broken into the parts Tri, gon, and metry, which means Three angle measurement, or equivalently Triangle measurement. Throughout this unit, we will
More informationUsing the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º.
Using the Quadrant Eye Piece Protractor Handle You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements 90º. Plumb Bob ø
More informationWORK SCHEDULE: MATHEMATICS 2007
, K WORK SCHEDULE: MATHEMATICS 00 GRADE MODULE TERM... LO NUMBERS, OPERATIONS AND RELATIONSHIPS able to recognise, represent numbers and their relationships, and to count, estimate, calculate and check
More informationMathematics (Project Maths Phase 1)
2011. S133S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination Sample Paper Mathematics (Project Maths Phase 1) Paper 2 Ordinary Level Time: 2 hours 300 marks Running
More informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8  Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationGeometry Enduring Understandings Students will understand 1. that all circles are similar.
High School  Circles Essential Questions: 1. Why are geometry and geometric figures relevant and important? 2. How can geometric ideas be communicated using a variety of representations? ******(i.e maps,
More informationMathematics Geometry Unit 1 (SAMPLE)
Review the Geometry sample yearlong scope and sequence associated with this unit plan. Mathematics Possible time frame: Unit 1: Introduction to Geometric Concepts, Construction, and Proof 14 days This
More informationWednesday 15 January 2014 Morning Time: 2 hours
Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Wednesday 15 January 2014 Morning Time: 2 hours Candidate Number
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, January 29, 2014 9:15 a.m. to 12:15 p.m., only Student Name: School Name: The possession or use of any
More informationAlgebra Geometry Glossary. 90 angle
lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:
More informationPreAlgebra 2008. Academic Content Standards Grade Eight Ohio. Number, Number Sense and Operations Standard. Number and Number Systems
Academic Content Standards Grade Eight Ohio PreAlgebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
More informationGeometry and Measurement
The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for
More informationGeometry. Higher Mathematics Courses 69. Geometry
The fundamental purpose of the course is to formalize and extend students geometric experiences from the middle grades. This course includes standards from the conceptual categories of and Statistics and
More informationBar Graphs with Intervals Grade Three
Bar Graphs with Intervals Grade Three Ohio Standards Connection Data Analysis and Probability Benchmark D Read, interpret and construct graphs in which icons represent more than a single unit or intervals
More informationExplorations with Shapes Kindergarten
Ohio Standards Connections Geometry and Spatial Sense Benchmark C Sort and compare twodimensional figures and threedimensional objects according to their characteristics and properties. Indicator 1 Identify
More informationTriangle Trigonometry and Circles
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
More informationComparing Sets of Data Grade Eight
Ohio Standards Connection: Data Analysis and Probability Benchmark C Compare the characteristics of the mean, median, and mode for a given set of data, and explain which measure of center best represents
More informationGeometry Course Summary Department: Math. Semester 1
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationGeometry 1. Unit 3: Perpendicular and Parallel Lines
Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples
More informationMain Idea in Informational Text Grade Three
Ohio Standards Connection Informational, Technical and Persuasive Text Benchmark C Identify the central ideas and supporting details of informational text. Indicator 3 Identify and list the important central
More informationMathematics. GCSE subject content and assessment objectives
Mathematics GCSE subject content and assessment objectives June 2013 Contents Introduction 3 Subject content 4 Assessment objectives 11 Appendix: Mathematical formulae 12 2 Introduction GCSE subject criteria
More informationPerformance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will
Performance Based Learning and Assessment Task Triangles in Parallelograms I. ASSESSSMENT TASK OVERVIEW & PURPOSE: In this task, students will discover and prove the relationship between the triangles
More informationTrigonometric Ratios TEACHER NOTES. About the Lesson. Vocabulary. Teacher Preparation and Notes. Activity Materials
About the Lesson In this activity, students discover the trigonometric ratios through measuring the side lengths of similar triangles and calculating their ratios. The formal definitions of the sine, cosine,
More informationDear Grade 4 Families,
Dear Grade 4 Families, During the next few weeks, our class will be exploring geometry. Through daily activities, we will explore the relationship between flat, twodimensional figures and solid, threedimensional
More informationPROVINCE OF THE EASTERN CAPE EDUCATION
PROVINCE OF THE EASTERN CAPE EDUCATION DIRECTORATE: CURRICULUM FET PROGRAMMES LESSON PLANS TERM 3 MATHEMATICS GRADE 10 FOREWORD The following Grade 10, 11 and 12 Lesson Plans were developed by Subject
More informationConjectures. Chapter 2. Chapter 3
Conjectures Chapter 2 C1 Linear Pair Conjecture If two angles form a linear pair, then the measures of the angles add up to 180. (Lesson 2.5) C2 Vertical Angles Conjecture If two angles are vertical
More informationKEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 447) Surveys and
More informationPerformance Assessment Task Which Shape? Grade 3. Common Core State Standards Math  Content Standards
Performance Assessment Task Which Shape? Grade 3 This task challenges a student to use knowledge of geometrical attributes (such as angle size, number of angles, number of sides, and parallel sides) to
More informationGrade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
More informationLesson 33: Example 1 (5 minutes)
Student Outcomes Students understand that the Law of Sines can be used to find missing side lengths in a triangle when you know the measures of the angles and one side length. Students understand that
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA. Thursday, August 16, 2012 8:30 to 11:30 a.m.
INTEGRATED ALGEBRA The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION INTEGRATED ALGEBRA Thursday, August 16, 2012 8:30 to 11:30 a.m., only Student Name: School Name: Print your name
More informationTrigonometry. An easy way to remember trigonometric properties is:
Trigonometry It is possible to solve many force and velocity problems by drawing vector diagrams. However, the degree of accuracy is dependent upon the exactness of the person doing the drawing and measuring.
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfdprep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More informationSample Test Questions
mathematics College Algebra Geometry Trigonometry Sample Test Questions A Guide for Students and Parents act.org/compass Note to Students Welcome to the ACT Compass Sample Mathematics Test! You are about
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More informationDear Accelerated PreCalculus Student:
Dear Accelerated PreCalculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, collegepreparatory mathematics course that will also
More informationUrbanization Grade Nine
Ohio Standards Connection: Geography Benchmark B Analyze geographic changes brought about by human activity using appropriate maps and other geographical data. Indicator 4 Explain the causes and consequences
More informationInv 1 5. Draw 2 different shapes, each with an area of 15 square units and perimeter of 16 units.
Covering and Surrounding: Homework Examples from ACE Investigation 1: Questions 5, 8, 21 Investigation 2: Questions 6, 7, 11, 27 Investigation 3: Questions 6, 8, 11 Investigation 5: Questions 15, 26 ACE
More informationThursday 28 February 2013 Afternoon
H Thursday 28 February 2013 Afternoon GCSE MATHEMATICS B J567/03 Paper 3 (Higher Tier) *J533610313* Candidates answer on the Question Paper. OCR supplied materials: None Other materials required: Geometrical
More informationRight Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring
Page 1 9 Trigonometry of Right Triangles Right Triangles A right triangle, as the one shown in Figure 5, is a triangle that has one angle measuring 90. The side opposite to the right angle is the longest
More informationHigh School Geometry Test Sampler Math Common Core Sampler Test
High School Geometry Test Sampler Math Common Core Sampler Test Our High School Geometry sampler covers the twenty most common questions that we see targeted for this level. For complete tests and break
More informationTime needed. Before the lesson Assessment task:
Formative Assessment Lesson Materials Alpha Version Beads Under the Cloud Mathematical goals This lesson unit is intended to help you assess how well students are able to identify patterns (both linear
More informationScience Rocks Grade Six
Ohio Standards Connections: Earth and Space Sciences Benchmark D Identify that the lithosphere contains rocks and minerals and that minerals make up rocks. Describe how rocks and minerals are formed and/or
More informationSystems of Transportation and Communication Grade Three
1 Ohio Standards Connection: Geography Benchmark D Analyze ways that transportation and communication relate to patterns of settlement and economic activity. Indicator 8 Identify systems of transportation
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationDrawing 3D Objects in Perspective
Mathematics Instructional Materials SAS#.1 (one per pair of students) SAS#.2 (one per pair of students) TIS#.1 (transparency) TIS#.2 (transparency) TIS#.3 (Journal prompt) Isometric Dot Paper Isometric
More informationTrigonometry for AC circuits
Trigonometry for AC circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More information7.4A/7.4B STUDENT ACTIVITY #1
7.4A/7.4B STUDENT ACTIVITY #1 Write a formula that could be used to find the radius of a circle, r, given the circumference of the circle, C. The formula in the Grade 7 Mathematics Chart that relates the
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationMathematics programmes of study: key stage 4. National curriculum in England
Mathematics programmes of study: key stage 4 National curriculum in England July 2014 Contents Purpose of study 3 Aims 3 Information and communication technology (ICT) 4 Spoken language 4 Working mathematically
More informationWhat s My Point?  Grade Six
Ohio Standards Connection Reading Applications: Informational, Technical and Persuasive Text Benchmark D Identify arguments and persuasive techniques used in persuasive writing. Indicators 6 Identify an
More informationConjectures for Geometry for Math 70 By I. L. Tse
Conjectures for Geometry for Math 70 By I. L. Tse Chapter Conjectures 1. Linear Pair Conjecture: If two angles form a linear pair, then the measure of the angles add up to 180. Vertical Angle Conjecture:
More informationEDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES. Maths Level 2. Chapter 5. Shape and space
Shape and space 5 EDEXCEL FUNCTIONAL SKILLS PILOT TEACHER S NOTES Maths Level 2 Chapter 5 Shape and space SECTION H 1 Perimeter 2 Area 3 Volume 4 2D Representations of 3D Objects 5 Remember what you
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel International GCSE Mathematics A Paper 1FR Centre Number Wednesday 14 May 2014 Morning Time: 2 hours Candidate Number Foundation Tier Paper Reference
More informationAngles and Quadrants. Angle Relationships and Degree Measurement. Chapter 7: Trigonometry
Chapter 7: Trigonometry Trigonometry is the study of angles and how they can be used as a means of indirect measurement, that is, the measurement of a distance where it is not practical or even possible
More informationFOREWORD. Executive Secretary
FOREWORD The Botswana Examinations Council is pleased to authorise the publication of the revised assessment procedures for the Junior Certificate Examination programme. According to the Revised National
More information2nd Semester Geometry Final Exam Review
Class: Date: 2nd Semester Geometry Final Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of an amusement park created a circular
More informationWEDNESDAY, 2 MAY 10.40 AM 11.15 AM. Date of birth Day Month Year Scottish candidate number
FOR OFFICIAL USE G KU RE Paper 1 Paper 2 2500/29/01 Total NATIONAL QUALIFICATIONS 2012 WEDNESDAY, 2 MAY 10.40 AM 11.15 AM MATHEMATICS STANDARD GRADE General Level Paper 1 Noncalculator Fill in these boxes
More informationMeasures of Spread and Their Effects Grade Seven
Ohio Standards Connection Data Analysis and Probability Benchmark F Determine and use the range, mean, median and mode to analyze and compare data, and explain what each indicates about the data. Indicator
More informationalternate interior angles
alternate interior angles two nonadjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
More informationPage. Trigonometry Sine Law and Cosine Law. push
Trigonometry Sine Law and Cosine Law Page Trigonometry can be used to calculate the side lengths and angle measures of triangles. Triangular shapes are used in construction to create rigid structures.
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B. Thursday, January 29, 2004 9:15 a.m. to 12:15 p.m.
The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION MATHEMATICS B Thursday, January 9, 004 9:15 a.m. to 1:15 p.m., only Print Your Name: Print Your School s Name: Print your name and
More informationVolume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms.
Volume of Right Prisms Objective To provide experiences with using a formula for the volume of right prisms. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game
More information